The Center of Mass

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The Center of Mass
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Finding the Center of Massby Integration
(Omit)
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Motion of the Center of Mass
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Center of Mass Motion
Definitions of center of mass motion.
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Conservation of Linear Momentum
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Kinetic Energy of a System
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System Kinetic Energy
(Eq. 8-18 frequently used in rotational dynamics.)
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Practice Momentum and Kinetic Energy
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• 08-2. Two masses move on a frictionless
  horizontal surface.
• M1 1kg, v1i 4m/s. M2 2kg, v2i 1m/s.
• Find the center of mass speed.

b) The masses collide along a straight line. Find
v1f if v2f 2.3 m/s and no other external
forces act.
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c) Calculate the initial and final kinetic
energies. Is the collision energetically
possible?
It is possible for kinetic energy to decrease due
to the production of thermal energy in a
collision.
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Collisions and Impulse
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Impulse is Area under F(t)
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Types of Collisions
?Complete Inelastic K ? Thermal
(v1f v2f) ? Inelastic K
? Thermal (v1f ? v2f) ? Elastic
Ki Kf (v1f ?
v2f)
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The Center-of-MassReference Frame
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Problems
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Practice Collisions and Impulse
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08-4. A bullet of 230 grains moves horizontally
at 830 feet per second and strikes a 10lb wood
block lying at rest on a horizontal surface. The
bullet takes 1.0 millisecond to stop inside the
block.
a) Convert the data to SI units.
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08-4 b) Calculate the speed the block moves at
just after the bullet stops in the block.
System momentum conserved when external impulse
is negligible.
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08-4 c) Calculate the kinetic energy of the
bullet before the collision and of the moving
block bullet after the collision. What percent
of the original kinetic energy is converted to
other energies? What percent is retained as
kinetic?
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08-4 d) Calculate the impulse received by the
block.
e) If the collision lasts 1.0 millisecond,
calculate the average force exerted on each
object.
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Practice
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08-5. Two masses move on a frictionless
horizontal surface. M1 1kg, v1i 4m/s. M2
4kg, v2i 1m/s in a laboratory. The masses
collide elastically along a straight line.
a) Show that in the center of mass frame that the
initial velocities are 2m/s and -1m/s.
(vcm 2 from previous example)
b) What are the final velocities in the lab frame?
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08-5 c) Calculate the system-momentum before and
after the collision in the lab-frame.
d) Calculate the initial and final kinetic
energies of the system in the lab-frame. Are
these energies consistent with the definition of
an elastic collision?
This is consistent with an elastic collision
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08-3. Two objects collide in two dimensions. No
external forces act at any time. In SI units
p1i (3, 4) p2i (2, 1)
a) If p1f (3, 2), then calculate p2f.
b) Make a sketch of the momentum vectors before
and after the collision.
a) Pi (3, 4) (2, 1) (3, 2) (px, py)
Pf (5, 5) (3px, 2py) px 2 py 3
p2f (2, 3)
Pi
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c) Calculate the angle each momentum vector makes
with the x-axis.
d) Angle of final total momentum vector Pf Pi
(5, 5)
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